Luca Cardelli

Luca Cardelli, Mirco Tribastone, Max Tschaikowski

Electric circuits manipulate electric charge and magnetic flux via a small set of discrete components to implement useful functionality over continuous time-varying signals represented by currents and voltages. Much of the same functionality is useful to biological organisms, where it is implemented by a completely different set of discrete components (typically proteins) and signal representations (typically via concentrations). We describe how to take a linear electric circuit and systematically convert it to a chemical reaction network of the same functionality, as a dynamical system. Both the structure and the components of the electric circuit are dissolved in the process, but the resulting chemical network is intelligible. This approach provides access to a large library of well-studied devices, from analog electronics, whose chemical network realization can be compared to natural biochemical networks, or used to engineer synthetic biochemical networks.

Luca Cardelli, Mirco Tribastone, Max Tschaikowski et al.

It is well known that exact notions of model abstraction and reduction for dynamical systems may not be robust enough in practice because they are highly sensitive to the specific choice of parameters. In this paper we consider this problem for nonlinear ordinary differential equations (ODEs) with polynomial derivatives. We introduce approximate differential equivalence as a more permissive variant of a recently developed exact counterpart, allowing ODE variables to be related even when they are governed by nearby derivatives. We develop algorithms to (i) compute the largest approximate differential equivalence; (ii) construct an approximate quotient model from the original one via an appropriate parameter perturbation; and (iii) provide a formal certificate on the quality of the approximation as an error bound, computed as an over-approximation of the reachable set of the perturbed model. Finally, we apply approximate differential equivalences to study the effect of parametric tolerances in models of symmetric electric circuits.

Max Whitby, Luca Cardelli, Marta Kwiatkowska et al.

Principles of feedback control have been shown to naturally arise in biological systems and successfully applied to build synthetic circuits. In this work we consider Biochemical Reaction Networks (CRNs) as a paradigm for modelling biochemical systems and provide the first implementation of a derivative component in CRNs. That is, given an input signal represented by the concentration level of some species, we build a CRN that produces as output the concentration of two species whose difference is the derivative of the input signal. By relying on this component, we present a CRN implementation of a feedback control loop with Proportional-Integral-Derivative (PID) controller and apply the resulting control architecture to regulate the protein expression in a microRNA regulated gene expression model.

Andrea Patane, Arno Blaas, Luca Laurenti et al.

Gaussian processes (GPs) enable principled computation of model uncertainty, making them attractive for safety-critical applications. Such scenarios demand that GP decisions are not only accurate, but also robust to perturbations. In this paper we present a framework to analyse adversarial robustness of GPs, defined as invariance of the model's decision to bounded perturbations. Given a compact subset of the input space $T\subseteq \mathbb{R}^d$, a point $x^*$ and a GP, we provide provable guarantees of adversarial robustness of the GP by computing lower and upper bounds on its prediction range in $T$. We develop a branch-and-bound scheme to refine the bounds and show, for any $ε> 0$, that our algorithm is guaranteed to converge to values $ε$-close to the actual values in finitely many iterations. The algorithm is anytime and can handle both regression and classification tasks, with analytical formulation for most kernels used in practice. We evaluate our methods on a collection of synthetic and standard benchmark datasets, including SPAM, MNIST and FashionMNIST. We study the effect of approximate inference techniques on robustness and demonstrate how our method can be used for interpretability. Our empirical results suggest that the adversarial robustness of GPs increases with accurate posterior estimation.

Kyriakos Polymenakos, Luca Laurenti, Andrea Patane et al.

Gaussian Processes (GPs) are widely employed in control and learning because of their principled treatment of uncertainty. However, tracking uncertainty for iterative, multi-step predictions in general leads to an analytically intractable problem. While approximation methods exist, they do not come with guarantees, making it difficult to estimate their reliability and to trust their predictions. In this work, we derive formal probability error bounds for iterative prediction and planning with GPs. Building on GP properties, we bound the probability that random trajectories lie in specific regions around the predicted values. Namely, given a tolerance $ε> 0 $, we compute regions around the predicted trajectory values, such that GP trajectories are guaranteed to lie inside them with probability at least $1-ε$. We verify experimentally that our method tracks the predictive uncertainty correctly, even when current approximation techniques fail. Furthermore, we show how the proposed bounds can be employed within a safe reinforcement learning framework to verify the safety of candidate control policies, guiding the synthesis of provably safe controllers.

Rhiannon Michelmore, Matthew Wicker, Luca Laurenti et al.

Deep neural network controllers for autonomous driving have recently benefited from significant performance improvements, and have begun deployment in the real world. Prior to their widespread adoption, safety guarantees are needed on the controller behaviour that properly take account of the uncertainty within the model as well as sensor noise. Bayesian neural networks, which assume a prior over the weights, have been shown capable of producing such uncertainty measures, but properties surrounding their safety have not yet been quantified for use in autonomous driving scenarios. In this paper, we develop a framework based on a state-of-the-art simulator for evaluating end-to-end Bayesian controllers. In addition to computing pointwise uncertainty measures that can be computed in real time and with statistical guarantees, we also provide a method for estimating the probability that, given a scenario, the controller keeps the car safe within a finite horizon. We experimentally evaluate the quality of uncertainty computation by several Bayesian inference methods in different scenarios and show how the uncertainty measures can be combined and calibrated for use in collision avoidance. Our results suggest that uncertainty estimates can greatly aid decision making in autonomous driving.

Arno Blaas, Andrea Patane, Luca Laurenti et al.

We investigate adversarial robustness of Gaussian Process Classification (GPC) models. Given a compact subset of the input space $T\subseteq \mathbb{R}^d$ enclosing a test point $x^*$ and a GPC trained on a dataset $\mathcal{D}$, we aim to compute the minimum and the maximum classification probability for the GPC over all the points in $T$. In order to do so, we show how functions lower- and upper-bounding the GPC output in $T$ can be derived, and implement those in a branch and bound optimisation algorithm. For any error threshold $ε> 0$ selected a priori, we show that our algorithm is guaranteed to reach values $ε$-close to the actual values in finitely many iterations. We apply our method to investigate the robustness of GPC models on a 2D synthetic dataset, the SPAM dataset and a subset of the MNIST dataset, providing comparisons of different GPC training techniques, and show how our method can be used for interpretability analysis. Our empirical analysis suggests that GPC robustness increases with more accurate posterior estimation.

Luca Cardelli, Marta Kwiatkowska, Luca Laurenti et al.

We introduce a probabilistic robustness measure for Bayesian Neural Networks (BNNs), defined as the probability that, given a test point, there exists a point within a bounded set such that the BNN prediction differs between the two. Such a measure can be used, for instance, to quantify the probability of the existence of adversarial examples. Building on statistical verification techniques for probabilistic models, we develop a framework that allows us to estimate probabilistic robustness for a BNN with statistical guarantees, i.e., with a priori error and confidence bounds. We provide experimental comparison for several approximate BNN inference techniques on image classification tasks associated to MNIST and a two-class subset of the GTSRB dataset. Our results enable quantification of uncertainty of BNN predictions in adversarial settings.

Luca Cardelli, Marta Kwiatkowska, Luca Laurenti et al.

Bayesian inference and Gaussian processes are widely used in applications ranging from robotics and control to biological systems. Many of these applications are safety-critical and require a characterization of the uncertainty associated with the learning model and formal guarantees on its predictions. In this paper we define a robustness measure for Bayesian inference against input perturbations, given by the probability that, for a test point and a compact set in the input space containing the test point, the prediction of the learning model will remain $δ-$close for all the points in the set, for $δ>0.$ Such measures can be used to provide formal guarantees for the absence of adversarial examples. By employing the theory of Gaussian processes, we derive tight upper bounds on the resulting robustness by utilising the Borell-TIS inequality, and propose algorithms for their computation. We evaluate our techniques on two examples, a GP regression problem and a fully-connected deep neural network, where we rely on weak convergence to GPs to study adversarial examples on the MNIST dataset.